Estimating the state of dynamically evolving systems is a fundamental challenge across diverse fields such as robotics, navigation, economics, and environmental monitoring. This thesis explores and compares three prominent state estimation methods: the Kalman Filter (KF), the Extended Kalman Filter (EKF), and the Unscented Kalman Filter (UKF), each tailored to handle specific complexities encountered in real-world applications.

The foundational Kalman Filter is rigorously examined first, deriving its algorithm through Bayesian inference and the fusion of multiple estimates. A comparative analysis of these approaches highlights the KF’s robustness in linear systems while acknowledging limitations in nonlinear environments.

The thesis then transitions to the Extended Kalman Filter, which extends the KF to nonlinear systems by linearizing state equations. Detailed mathematical derivation and comparative studies underscore the EKF’s enhanced capabilities in handling complex dynamics, yet reveal challenges in accuracy and computational cost.

Moving further, the Unscented Kalman Filter is introduced as a non-linear state estimation method utilizing the Unscented Transform. Detailed exploration and mathematical formulation demonstrate its effectiveness in addressing uncertainties, presenting a viable alternative to both KF and EKF in scenarios where linearization proves inadequate.

To validate these methodologies, simulations are conducted using real-world data from the KITTI dataset, comprising of GPS and IMU measurements. Ground truth trajectories and non-linear variables such as yaw rates and forward velocities are utilized, showcasing each filter’s ability to estimate and track dynamic system states accurately.

Results from simulations are analyzed using performance metrics including Normalized Estimation Error Squared (NEES) and Root Mean Squared Error (RMSE), providing quantitative insights into filter performance relative to ground truth. These evaluations emphasize the strengths and limitations of each method across various application domains, supporting informed decisions on filter selection based on specific system dynamics and measurement characteristics.

In conclusion, this thesis contributes to a comprehensive analysis and comparative study of state estimation methods essential for navigating the complexities of dynamically evolving systems. By bridging theoretical advancements with practical insights, it lays a foundation for future research and application in fields requiring precise state estimation amidst dynamic change.

The Kalman filter is a recursive algorithm that estimates the state of a dynamic system subject to measurement and model noise. If all noise terms affecting the system are white Gaussian noise with known mean and variance, and all noise terms are independent of each other, then the Kalman filter is the optimal estimator for the state variable. When measurements are collected from multiple sources, the covariance between these sources should be known or the sources should be independent to ensure that the estimate made by the Kalman filter is optimal. When the covariance between dependent measurement sources is not known, various methods exist which provide a solution to this problem. This thesis discusses two methods: the H∞ filter and covariance intersection...

The Curie-Weiss model is a simplification of the Ising model to show the existence of a phase transition for ferromagnetism. In this thesis, we study the behaviour of sums of these dependent variables. We prove in general that under the appropriate assumptions, we can still conclude a version of the Law of Large Numbers. We also find that if there exists a certain m∈R, λ>0 and integer k≥1, we have that (S_n-nm)/n^1/2k converges to exp(-λs^2k/(2k)!) in distribution.
For the Curie-Weiss model this means that for β, which is a constant proportion to inverse temperature, we find that if β∈(0,1) we have S_n/n→δ(s) and S_n/√n→ N(0,σ²) in distribution where σ²=(1-β)^-1-1. At β=1 there occurs a phase transition, we still have that S_n/n→δ(s), but now S_n/n^3/4→\exp(-s⁴/12). When β>1 we can find an m>0 such that S_n/n→½[δ(s-m)+δ(s+m)].
We also study the Curie-Weiss model where we assume that it is under the influence of a magnetic field. We prove that we do not find a phase transition, and we always have S_n/n→δ(s-m) in distribution for some m∈R. Next to this we find that (S_n-nm)/√n always converges to a normal distribution.

In this bachelor thesis we use a stochastic model to aspire to explain biodiversity patterns in different ecosystems with selection advantage. The stochastic model we use is an extension of the mean­-field voter model where we include a selection factor. In the model individuals with two different types of alleles in two different ecosystems are considered. The model is a stochastic Markov process that describes interactions of individuals with each­other over time. This means that the ratio of individuals with certain alleles stochastically drifts over time. The main goal of this bachelor thesis is investigate whether is it possible that individuals with two different types of alleles can coexist (a stable equi­librium) in two populations. We do this by taking the limit of this Markov process such that we can show convergence to ordinary differential equations. By studying these differential equations we ob­tain results: vector fields with equilibrium points. We conclude that, under certain conditions and with a selection advantage, coexistence of individuals with two different alleles in two different ecosystems is possible (see Section 4.4) in the form of a stable equilibrium. Furthermore we claim that the typical time to absorption, reaching an absorbing state where all the individuals have the same allele, of the two­-dimensional mean­-field voter model with selection scales exponentially with the system size.

We inspect the behavior of the probability that a weighted sum of random variables with log-normal tails is greater than its expected value. Under the right conditions for the weights and the variance being set to 1; we were able to bound a suitable transformation of this probability with the upper bound being a fixed factor of the square root of e above the lower bound. Beyond this, we analyse the conditions on the weights and determine a method for letting the weights be random and give an example.
We end off by extending our result to general variance, where we see that the deviation between the lower and upper bound as well as the domain for the result are dependant on the variance.

As an insurer you want identify the risks you take to prevent bankruptcy. The
theory of large deviations formalizes the study of such rare events. We will use the
theorem of Cramér, which is a main theorem in large deviation theory, to investigate
the rate at which the probability of large deviations of the sums of random variables
decay. Using Sanov’s theorem we will derive an expression for large deviations of
the empirical measure. Furthermore, we will use Gibbs’s principle to derive the
distribution of random variables conditional on a large deviation.